Linear models — distributions of the regression sums of squares

Let $Y=X\beta+\epsilon$ be a multiple linear regression model with $p$ regressors and an intercept, where $\beta=(\beta_0,\beta_1,…,\beta_p)^{T}$ and the random error $\epsilon\sim N_n(0,\sigma^2I)$, $\sigma>0$ and $n>p+1$. Here, the least squares estimation method provides unique estimator of $\beta$, say $\hat\beta$. Let the total sum of squares (corrected), sum of squares due to regression/model and sum of squares due to error/residual, based on $\hat\beta$, be denoted by $Y^{T}AY$, $Y^{T}BY$ and $Y^{T}CY$, respectively, so that $Y^{T}AY=Y^{T}BY+Y^{T}CY$. Which of the following statements are always true?